Weighted sampling -- sampling from a probability density function (PDF) proportional to the product of a base PDF and a weight function -- is a fundamental technique with wide-ranging applications in variance reduction, biased sampling, data augmentation, and more. Leveraging the increasing availability of pretrained score-based generative models (SGMs), we propose a training-free weighted sampling framework that approximates the backward diffusion process of the target distribution by augmenting the pretrained base score function with an auxiliary guidance term, in a principled and computationally efficient manner. Our approach builds on two key components: a lightweight approximation of the guidance that avoids costly higher-order derivatives of both the score and weight functions, and an uncertainty-aware scheduler that dynamically adjusts the guidance strength based on a temporal analysis of approximation error. Together, these components enable accurate and stable sampling without relying on particle-based resampling or Hessian evaluations commonly required by existing methods. We validate the effectiveness of our method from synthetic to large-scale settings such as Stable Diffusion XL, where our framework achieves $1.2\times$ to $4.7\times$ speedups while consistently matching or outperforming state-of-the-art baselines in task performance. These results position our method as a scalable and inference-efficient solution for task-adaptive, time-sensitive sampling in generative applications.